where \(n\)  is the number of observations, \(U_{i}^{(j)}\)  denotes the \(j\) -th pseudo-observation of the \(i\) -th random variable, with \(i=1,\dots ,d\) , and \(\mathbb {1}_A\)  is the indicator function.
However, strong complexity limitations occur for increasing values of \(d\) , forcing to either move towards other kernel cumulative estimators or towards a parametric family of multivariate copulas. For the latter, common models are the multivariate Gaussian copula with correlation matrix \(\Sigma \)  or the multivariate Student's t-copula with \(\nu \)  degrees of freedom and correlation matrix \(\Sigma \) , suitable for extreme value dependence [1]}.
Archimedean copulas assume the form \(C(\mathbf {u})=\phi ^{-1}(\phi (u_1)+\dots +\phi (u_d))\)  where \(\phi \)  is the generator function of the Archimedean copula. Such structure is said to be exchangeable, i.e., the components can be swapped indifferently, but its symmetry introduces modeling limitations. Multivariate copulas built using bivariate pair-copulas, also referred to as vine copulas, represent a more flexible model but the selection of the vine tree structure and the pair copula families is a complex task [2]}.
